Definition:Axiom/Formal Systems/Axiom Schema

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Definition

Let $\mathcal L$ be a formal language.

Part of defining a proof system $\mathscr P$ for $\mathcal L$ is to specify its axiom schemata.


An axiom schema is a well-formed formula $\phi$ of $\mathcal L$, except for it containing one or more variables which are outside $\mathcal L$ itself.

This formula can then be used to represent an infinite number of individual axioms in one statement.


Namely, each of these variables is allowed to take a specified range of values, most commonly WFFs.

Each WFF $\psi$ that results from $\phi$ by a valid choice of values for all the variables is then an axiom of $\mathscr P$.


Linguistic Note

The plural of axiom schema is correctly axiom schemata, but it is commonplace to see the word schemas used for schemata.


Examples

It was proved by Richard Merett Montague in 1957 that ZFC and Peano arithmetic require an axiom schema.


Sources